statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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iso_of_components (H : X.1 ≅ Y.1) (α : H.hom _* X.2 ≅ Y.2) : X ≅ Y | { hom := { base := H.hom, c := α.inv },
inv := { base := H.inv,
c := presheaf.to_pushforward_of_iso H α.hom },
hom_inv_id' := by { ext, { simp, erw category.id_comp, simpa }, simp },
inv_hom_id' :=
begin
ext x,
induction x using opposite.rec,
simp only [comp_c_app, whisker_right_app, presheaf.to... | def | algebraic_geometry.PresheafedSpace.iso_of_components | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"opposite.rec"
] | An isomorphism of PresheafedSpaces is a homeomorphism of the underlying space, and a
natural transformation between the sheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf_iso_of_iso (H : X ≅ Y) : Y.2 ≅ H.hom.base _* X.2 | { hom := H.hom.c,
inv := presheaf.pushforward_to_of_iso ((forget _).map_iso H).symm H.inv.c,
hom_inv_id' :=
begin
ext U,
have := congr_app H.inv_hom_id U,
simp only [comp_c_app, id_c_app,
eq_to_hom_map, eq_to_hom_trans] at this,
generalize_proofs h at this,
simpa using congr_arg (λ f, f ... | def | algebraic_geometry.PresheafedSpace.sheaf_iso_of_iso | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | Isomorphic PresheafedSpaces have natural isomorphic presheaves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_is_iso_of_iso (f : X ⟶ Y) [is_iso f] : is_iso f.base | is_iso.of_iso ((forget _).map_iso (as_iso f)) | instance | algebraic_geometry.PresheafedSpace.base_is_iso_of_iso | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c_is_iso_of_iso (f : X ⟶ Y) [is_iso f] : is_iso f.c | is_iso.of_iso (sheaf_iso_of_iso (as_iso f)) | instance | algebraic_geometry.PresheafedSpace.c_is_iso_of_iso | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_components (f : X ⟶ Y) [is_iso f.base] [is_iso f.c] : is_iso f | begin
convert is_iso.of_iso (iso_of_components (as_iso f.base) (as_iso f.c).symm),
ext, { simpa }, { simp },
end | lemma | algebraic_geometry.PresheafedSpace.is_iso_of_components | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | This could be used in conjunction with `category_theory.nat_iso.is_iso_of_is_iso_app`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict {U : Top} (X : PresheafedSpace.{v v u} C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) : PresheafedSpace C | { carrier := U,
presheaf := h.is_open_map.functor.op ⋙ X.presheaf } | def | algebraic_geometry.PresheafedSpace.restrict | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"Top",
"open_embedding"
] | The restriction of a presheafed space along an open embedding into the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_restrict {U : Top} (X : PresheafedSpace.{v v u} C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) :
X.restrict h ⟶ X | { base := f,
c := { app := λ V, X.presheaf.map (h.is_open_map.adjunction.counit.app V.unop).op,
naturality' := λ U V f, show _ = _ ≫ X.presheaf.map _,
by { rw [← map_comp, ← map_comp], refl } } } | def | algebraic_geometry.PresheafedSpace.of_restrict | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"Top",
"map_comp",
"open_embedding"
] | The map from the restriction of a presheafed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_restrict_mono {U : Top} (X : PresheafedSpace C) (f : U ⟶ X.1)
(hf : open_embedding f) : mono (X.of_restrict hf) | begin
haveI : mono f := (Top.mono_iff_injective _).mpr hf.inj,
constructor,
intros Z g₁ g₂ eq,
ext V,
{ induction V using opposite.rec,
have hV : (opens.map (X.of_restrict hf).base).obj (hf.is_open_map.functor.obj V) = V,
{ ext1, exact set.preimage_image_eq _ hf.inj },
haveI : is_iso (hf.i... | instance | algebraic_geometry.PresheafedSpace.of_restrict_mono | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"Top",
"Top.mono_iff_injective",
"open_embedding",
"opposite.rec",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_top_presheaf (X : PresheafedSpace C) :
(X.restrict (opens.open_embedding ⊤)).presheaf =
(opens.inclusion_top_iso X.carrier).inv _* X.presheaf | by { dsimp, rw opens.inclusion_top_functor X.carrier, refl } | lemma | algebraic_geometry.PresheafedSpace.restrict_top_presheaf | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_top_c (X : PresheafedSpace C) :
(X.of_restrict (opens.open_embedding ⊤)).c = eq_to_hom
(by { rw [restrict_top_presheaf, ←presheaf.pushforward.comp_eq],
erw iso.inv_hom_id, rw presheaf.pushforward.id_eq }) | /- another approach would be to prove the left hand side
is a natural isoomorphism, but I encountered a universe
issue when `apply nat_iso.is_iso_of_is_iso_app`. -/
begin
ext U, change X.presheaf.map _ = _, convert eq_to_hom_map _ _ using 1,
congr, simpa,
{ induction U using opposite.rec, dsimp, congr, ... | lemma | algebraic_geometry.PresheafedSpace.of_restrict_top_c | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_restrict_top (X : PresheafedSpace C) :
X ⟶ X.restrict (opens.open_embedding ⊤) | { base := (opens.inclusion_top_iso X.carrier).inv,
c := eq_to_hom (restrict_top_presheaf X) } | def | algebraic_geometry.PresheafedSpace.to_restrict_top | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | The map to the restriction of a presheafed space along the canonical inclusion from the top
subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_top_iso (X : PresheafedSpace C) :
X.restrict (opens.open_embedding ⊤) ≅ X | { hom := X.of_restrict _,
inv := X.to_restrict_top,
hom_inv_id' := ext _ _ (concrete_category.hom_ext _ _ $ λ ⟨x, _⟩, rfl) $
by { erw comp_c, rw X.of_restrict_top_c, ext, simp },
inv_hom_id' := ext _ _ rfl $
by { erw comp_c, rw X.of_restrict_top_c, ext, simpa [-eq_to_hom_refl] } } | def | algebraic_geometry.PresheafedSpace.restrict_top_iso | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | The isomorphism from the restriction to the top subspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : (PresheafedSpace.{v v u} C)ᵒᵖ ⥤ C | { obj := λ X, (unop X).presheaf.obj (op ⊤),
map := λ X Y f, f.unop.c.app (op ⊤) } | def | algebraic_geometry.PresheafedSpace.Γ | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | The global sections, notated Gamma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ_obj_op (X : PresheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.PresheafedSpace.Γ_obj_op | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map_op {X Y : PresheafedSpace.{v v u} C} (f : X ⟶ Y) :
Γ.map f.op = f.c.app (op ⊤) | rfl | lemma | algebraic_geometry.PresheafedSpace.Γ_map_op | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_presheaf (F : C ⥤ D) : PresheafedSpace.{v v u} C ⥤ PresheafedSpace.{v v u} D | { obj := λ X, { carrier := X.carrier, presheaf := X.presheaf ⋙ F },
map := λ X Y f, { base := f.base, c := whisker_right f.c F }, } | def | category_theory.functor.map_presheaf | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | We can apply a functor `F : C ⥤ D` to the values of the presheaf in any `PresheafedSpace C`,
giving a functor `PresheafedSpace C ⥤ PresheafedSpace D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_presheaf_obj_X (F : C ⥤ D) (X : PresheafedSpace C) :
((F.map_presheaf.obj X) : Top.{v}) = (X : Top.{v}) | rfl | lemma | category_theory.functor.map_presheaf_obj_X | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_presheaf_obj_presheaf (F : C ⥤ D) (X : PresheafedSpace C) :
(F.map_presheaf.obj X).presheaf = X.presheaf ⋙ F | rfl | lemma | category_theory.functor.map_presheaf_obj_presheaf | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_presheaf_map_f (F : C ⥤ D) {X Y : PresheafedSpace.{v v u} C} (f : X ⟶ Y) :
(F.map_presheaf.map f).base = f.base | rfl | lemma | category_theory.functor.map_presheaf_map_f | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_presheaf_map_c (F : C ⥤ D) {X Y : PresheafedSpace.{v v u} C} (f : X ⟶ Y) :
(F.map_presheaf.map f).c = whisker_right f.c F | rfl | lemma | category_theory.functor.map_presheaf_map_c | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
on_presheaf {F G : C ⥤ D} (α : F ⟶ G) : G.map_presheaf ⟶ F.map_presheaf | { app := λ X,
{ base := 𝟙 _,
c := whisker_left X.presheaf α ≫ eq_to_hom (presheaf.pushforward.id_eq _).symm } } | def | category_theory.nat_trans.on_presheaf | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | A natural transformation induces a natural transformation between the `map_presheaf` functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reduced : Prop | (component_reduced : ∀ U, _root_.is_reduced (X.presheaf.obj (op U)) . tactic.apply_instance) | class | algebraic_geometry.is_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_reduced"
] | A scheme `X` is reduced if all `𝒪ₓ(U)` are reduced. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reduced_of_stalk_is_reduced [∀ x : X.carrier, _root_.is_reduced (X.presheaf.stalk x)] :
is_reduced X | begin
refine ⟨λ U, ⟨λ s hs, _⟩⟩,
apply presheaf.section_ext X.sheaf U s 0,
intro x,
rw ring_hom.map_zero,
change X.presheaf.germ x s = 0,
exact (hs.map _).eq_zero
end | lemma | algebraic_geometry.is_reduced_of_stalk_is_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_reduced",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_is_reduced_of_reduced [is_reduced X] (x : X.carrier) :
_root_.is_reduced (X.presheaf.stalk x) | begin
constructor,
rintros g ⟨n, e⟩,
obtain ⟨U, hxU, s, rfl⟩ := X.presheaf.germ_exist x g,
rw [← map_pow, ← map_zero (X.presheaf.germ ⟨x, hxU⟩)] at e,
obtain ⟨V, hxV, iU, iV, e'⟩ := X.presheaf.germ_eq x hxU hxU _ 0 e,
rw [map_pow, map_zero] at e',
replace e' := (is_nilpotent.mk _ _ e').eq_zero,
erw ← co... | instance | algebraic_geometry.stalk_is_reduced_of_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_nilpotent.mk",
"is_reduced",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_of_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]
[is_reduced Y] : is_reduced X | begin
constructor,
intro U,
have : U = (opens.map f.1.base).obj (H.base_open.is_open_map.functor.obj U),
{ ext1, exact (set.preimage_image_eq _ H.base_open.inj).symm },
rw this,
exact is_reduced_of_injective (inv $ f.1.c.app (op $ H.base_open.is_open_map.functor.obj U))
(as_iso $ f.1.c.app (op $ H.base_... | lemma | algebraic_geometry.is_reduced_of_open_immersion | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_reduced",
"is_reduced_of_injective",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_is_reduced_iff (R : CommRing) :
is_reduced (Scheme.Spec.obj $ op R) ↔ _root_.is_reduced R | begin
refine ⟨_, λ h, by exactI infer_instance⟩,
intro h,
resetI,
haveI : _root_.is_reduced (LocallyRingedSpace.Γ.obj (op $ Spec.to_LocallyRingedSpace.obj $ op R)),
{ change _root_.is_reduced ((Scheme.Spec.obj $ op R).presheaf.obj $ op ⊤), apply_instance },
exact is_reduced_of_injective (to_Spec_Γ R)
((... | lemma | algebraic_geometry.affine_is_reduced_iff | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing",
"is_reduced",
"is_reduced_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_of_is_affine_is_reduced [is_affine X]
[h : _root_.is_reduced (X.presheaf.obj (op ⊤))] : is_reduced X | begin
haveI : is_reduced (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))),
{ rw affine_is_reduced_iff, exact h },
exact is_reduced_of_open_immersion X.iso_Spec.hom,
end | lemma | algebraic_geometry.is_reduced_of_is_affine_is_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reduce_to_affine_global (P : ∀ (X : Scheme) (U : opens X.carrier), Prop)
(h₁ : ∀ (X : Scheme) (U : opens X.carrier),
(∀ (x : U), ∃ {V} (h : x.1 ∈ V) (i : V ⟶ U), P X V) → P X U)
(h₂ : ∀ {X Y} (f : X ⟶ Y) [hf : is_open_immersion f], ∃ {U : set X.carrier} {V : set Y.carrier}
(hU : U = ⊤) (hV : V = set.range f... | begin
intros X U,
apply h₁,
intro x,
obtain ⟨_, ⟨j, rfl⟩, hx, i⟩ := X.affine_basis_cover_is_basis.exists_subset_of_mem_open
(set_like.mem_coe.2 x.prop) U.is_open,
let U' : opens _ := ⟨_, (X.affine_basis_cover.is_open j).base_open.open_range⟩,
let i' : U' ⟶ U :=
hom_of_le i,
refine ⟨U', hx, i', _⟩,... | lemma | algebraic_geometry.reduce_to_affine_global | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing",
"set.range"
] | To show that a statement `P` holds for all open subsets of all schemes, it suffices to show that
1. In any scheme `X`, if `P` holds for an open cover of `U`, then `P` holds for `U`.
2. For an open immerison `f : X ⟶ Y`, if `P` holds for the entire space of `X`, then `P` holds for
the image of `f`.
3. `P` holds for th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reduce_to_affine_nbhd (P : ∀ (X : Scheme) (x : X.carrier), Prop)
(h₁ : ∀ (R : CommRing) (x : prime_spectrum R), P (Scheme.Spec.obj $ op R) x)
(h₂ : ∀ {X Y} (f : X ⟶ Y) [is_open_immersion f] (x : X.carrier), P X x → P Y (f.1.base x)) :
∀ (X : Scheme) (x : X.carrier), P X x | begin
intros X x,
obtain ⟨y, e⟩ := X.affine_cover.covers x,
convert h₂ (X.affine_cover.map (X.affine_cover.f x)) y _,
{ rw e },
apply h₁,
end | lemma | algebraic_geometry.reduce_to_affine_nbhd | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_basic_open_eq_bot {X : Scheme} [hX : is_reduced X] {U : opens X.carrier}
(s : X.presheaf.obj (op U)) (hs : X.basic_open s = ⊥) :
s = 0 | begin
apply Top.presheaf.section_ext X.sheaf U,
simp_rw ring_hom.map_zero,
unfreezingI { revert X U hX s },
refine reduce_to_affine_global _ _ _ _,
{ intros X U hx hX s hs x,
obtain ⟨V, hx, i, H⟩ := hx x,
unfreezingI { specialize H (X.presheaf.map i.op s) },
erw Scheme.basic_open_res at H,
rw ... | lemma | algebraic_geometry.eq_zero_of_basic_open_eq_bot | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"Top.presheaf.section_ext",
"inf_bot_eq",
"is_nilpotent.eq_zero",
"is_reduced",
"prime_spectrum.basic_open_eq_bot_iff",
"ring_hom.map_zero",
"set.image_univ",
"set.preimage_image_eq",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_bot_iff {X : Scheme} [is_reduced X] {U : opens X.carrier}
(s : X.presheaf.obj $ op U) :
X.basic_open s = ⊥ ↔ s = 0 | begin
refine ⟨eq_zero_of_basic_open_eq_bot s, _⟩,
rintro rfl,
simp,
end | lemma | algebraic_geometry.basic_open_eq_bot_iff | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral : Prop | (nonempty : nonempty X.carrier . tactic.apply_instance)
(component_integral : ∀ (U : opens X.carrier) [_root_.nonempty U],
is_domain (X.presheaf.obj (op U)) . tactic.apply_instance) | class | algebraic_geometry.is_integral | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_domain",
"is_integral"
] | A scheme `X` is integral if its carrier is nonempty,
and `𝒪ₓ(U)` is an integral domain for each `U ≠ ∅`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reduced_of_is_integral [is_integral X] : is_reduced X | begin
constructor,
intro U,
cases U.1.eq_empty_or_nonempty with h h,
{ have : U = ⊥ := set_like.ext' h,
haveI := CommRing.subsingleton_of_is_terminal (X.sheaf.is_terminal_of_eq_empty this),
change _root_.is_reduced (X.sheaf.val.obj (op U)),
apply_instance },
{ haveI : nonempty U := by simpa, apply... | instance | algebraic_geometry.is_reduced_of_is_integral | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing.subsingleton_of_is_terminal",
"is_integral",
"is_reduced",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_of_is_integral [is_integral X] : irreducible_space X.carrier | begin
by_contradiction H,
replace H : ¬ is_preirreducible (⊤ : set X.carrier) := λ h,
H { to_preirreducible_space := ⟨h⟩, to_nonempty := infer_instance },
simp_rw [is_preirreducible_iff_closed_union_closed, not_forall, not_or_distrib] at H,
rcases H with ⟨S, T, hS, hT, h₁, h₂, h₃⟩,
erw not_forall at h₂ h₃... | instance | algebraic_geometry.is_irreducible_of_is_integral | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing.of",
"CommRing.prod_fan_is_limit",
"by_contradiction",
"false_of_nontrivial_of_product_domain",
"irreducible_space",
"is_integral",
"is_preirreducible",
"is_preirreducible_iff_closed_union_closed",
"not_forall",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_is_irreducible_is_reduced [is_reduced X] [H : irreducible_space X.carrier] :
is_integral X | begin
split, intros U hU,
haveI := (@@LocallyRingedSpace.component_nontrivial X.to_LocallyRingedSpace U hU).1,
haveI : no_zero_divisors
(X.to_LocallyRingedSpace.to_SheafedSpace.to_PresheafedSpace.presheaf.obj (op U)),
{ refine ⟨λ a b e, _⟩,
simp_rw [← basic_open_eq_bot_iff, ← opens.not_nonempty_iff_eq_b... | lemma | algebraic_geometry.is_integral_of_is_irreducible_is_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"irreducible_space",
"is_integral",
"is_reduced",
"no_zero_divisors",
"no_zero_divisors.to_is_domain",
"nonempty_preirreducible_inter",
"ring_hom.map_mul",
"ring_hom.map_zero",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_iff_is_irreducible_and_is_reduced :
is_integral X ↔ irreducible_space X.carrier ∧ is_reduced X | ⟨λ _, by exactI ⟨infer_instance, infer_instance⟩,
λ ⟨_, _⟩, by exactI is_integral_of_is_irreducible_is_reduced X⟩ | lemma | algebraic_geometry.is_integral_iff_is_irreducible_and_is_reduced | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"irreducible_space",
"is_integral",
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]
[is_integral Y] [nonempty X.carrier] : is_integral X | begin
constructor,
intros U hU,
have : U = (opens.map f.1.base).obj (H.base_open.is_open_map.functor.obj U),
{ ext1, exact (set.preimage_image_eq _ H.base_open.inj).symm },
rw this,
haveI : is_domain (Y.presheaf.obj (op (H.base_open.is_open_map.functor.obj U))),
{ apply_with is_integral.component_integral... | lemma | algebraic_geometry.is_integral_of_open_immersion | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_domain",
"is_integral",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_is_integral_iff (R : CommRing) :
is_integral (Scheme.Spec.obj $ op R) ↔ is_domain R | ⟨λ h, by exactI ring_equiv.is_domain ((Scheme.Spec.obj $ op R).presheaf.obj _)
(as_iso $ to_Spec_Γ R).CommRing_iso_to_ring_equiv, λ h, by exactI infer_instance⟩ | lemma | algebraic_geometry.affine_is_integral_iff | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"CommRing",
"is_domain",
"is_integral",
"ring_equiv.is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_is_affine_is_domain [is_affine X] [nonempty X.carrier]
[h : is_domain (X.presheaf.obj (op ⊤))] : is_integral X | begin
haveI : is_integral (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))),
{ rw affine_is_integral_iff, exact h },
exact is_integral_of_open_immersion X.iso_Spec.hom,
end | lemma | algebraic_geometry.is_integral_of_is_affine_is_domain | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_domain",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective_of_is_integral [is_integral X] {U V : opens X.carrier} (i : U ⟶ V)
[H : nonempty U] :
function.injective (X.presheaf.map i.op) | begin
rw injective_iff_map_eq_zero,
intros x hx,
rw ← basic_open_eq_bot_iff at ⊢ hx,
rw Scheme.basic_open_res at hx,
revert hx,
contrapose!,
simp_rw [← opens.not_nonempty_iff_eq_bot, not_not],
apply nonempty_preirreducible_inter U.is_open (RingedSpace.basic_open _ _).is_open,
simpa using H
end | lemma | algebraic_geometry.map_injective_of_is_integral | algebraic_geometry | src/algebraic_geometry/properties.lean | [
"algebraic_geometry.AffineScheme",
"ring_theory.nilpotent",
"topology.sheaves.sheaf_condition.sites",
"algebra.category.Ring.constructions",
"ring_theory.local_properties"
] | [
"is_integral",
"is_open",
"nonempty_preirreducible_inter",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V (i j : 𝒰.J) : Scheme | pullback ((pullback.fst : pullback ((𝒰.map i) ≫ f) g ⟶ _) ≫ (𝒰.map i)) (𝒰.map j) | def | algebraic_geometry.Scheme.pullback.V | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The intersection of `Uᵢ ×[Z] Y` and `Uⱼ ×[Z] Y` is given by (Uᵢ ×[Z] Y) ×[X] Uⱼ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t (i j : 𝒰.J) : V 𝒰 f g i j ⟶ V 𝒰 f g j i | begin
haveI : has_pullback (pullback.snd ≫ 𝒰.map i ≫ f) g :=
has_pullback_assoc_symm (𝒰.map j) (𝒰.map i) (𝒰.map i ≫ f) g,
haveI : has_pullback (pullback.snd ≫ 𝒰.map j ≫ f) g :=
has_pullback_assoc_symm (𝒰.map i) (𝒰.map j) (𝒰.map j ≫ f) g,
refine (pullback_symmetry _ _).hom ≫ _,
refine (pullback_a... | def | algebraic_geometry.Scheme.pullback.t | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The canonical transition map `(Uᵢ ×[Z] Y) ×[X] Uⱼ ⟶ (Uⱼ ×[Z] Y) ×[X] Uᵢ` given by the fact
that pullbacks are associative and symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t_fst_fst (i j : 𝒰.J) : t 𝒰 f g i j ≫ pullback.fst ≫ pullback.fst = pullback.snd | begin
delta t,
simp only [category.assoc, id.def, pullback_symmetry_hom_comp_fst_assoc,
pullback_assoc_hom_snd_fst, pullback.lift_fst_assoc, pullback_symmetry_hom_comp_snd,
pullback_assoc_inv_fst_fst, pullback_symmetry_hom_comp_fst],
end | lemma | algebraic_geometry.Scheme.pullback.t_fst_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_fst_snd (i j : 𝒰.J) :
t 𝒰 f g i j ≫ pullback.fst ≫ pullback.snd = pullback.fst ≫ pullback.snd | begin
delta t,
simp only [pullback_symmetry_hom_comp_snd_assoc, category.comp_id, category.assoc, id.def,
pullback_symmetry_hom_comp_fst_assoc, pullback_assoc_hom_snd_snd, pullback.lift_snd,
pullback_assoc_inv_snd],
end | lemma | algebraic_geometry.Scheme.pullback.t_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_snd (i j : 𝒰.J) :
t 𝒰 f g i j ≫ pullback.snd = pullback.fst ≫ pullback.fst | begin
delta t,
simp only [pullback_symmetry_hom_comp_snd_assoc, category.assoc, id.def,
pullback_symmetry_hom_comp_snd, pullback_assoc_hom_fst, pullback.lift_fst_assoc,
pullback_symmetry_hom_comp_fst, pullback_assoc_inv_fst_snd],
end | lemma | algebraic_geometry.Scheme.pullback.t_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t_id (i : 𝒰.J) : t 𝒰 f g i i = 𝟙 _ | begin
apply pullback.hom_ext; rw category.id_comp,
apply pullback.hom_ext,
{ rw ← cancel_mono (𝒰.map i), simp only [pullback.condition, category.assoc, t_fst_fst] },
{ simp only [category.assoc, t_fst_snd]},
{ rw ← cancel_mono (𝒰.map i),simp only [pullback.condition, t_snd, category.assoc] }
end | lemma | algebraic_geometry.Scheme.pullback.t_id | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fV (i j : 𝒰.J) : V 𝒰 f g i j ⟶ pullback ((𝒰.map i) ≫ f) g | pullback.fst | abbreviation | algebraic_geometry.Scheme.pullback.fV | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The inclusion map of `V i j = (Uᵢ ×[Z] Y) ×[X] Uⱼ ⟶ Uᵢ ×[Z] Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t' (i j k : 𝒰.J) :
pullback (fV 𝒰 f g i j) (fV 𝒰 f g i k) ⟶ pullback (fV 𝒰 f g j k) (fV 𝒰 f g j i) | begin
refine (pullback_right_pullback_fst_iso _ _ _).hom ≫ _,
refine _ ≫ (pullback_symmetry _ _).hom,
refine _ ≫ (pullback_right_pullback_fst_iso _ _ _).inv,
refine pullback.map _ _ _ _ (t 𝒰 f g i j) (𝟙 _) (𝟙 _) _ _,
{ simp only [←pullback.condition, category.comp_id, t_fst_fst_assoc] },
{ simp only [cat... | def | algebraic_geometry.Scheme.pullback.t' | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The map `((Xᵢ ×[Z] Y) ×[X] Xⱼ) ×[Xᵢ ×[Z] Y] ((Xᵢ ×[Z] Y) ×[X] Xₖ)` ⟶
`((Xⱼ ×[Z] Y) ×[X] Xₖ) ×[Xⱼ ×[Z] Y] ((Xⱼ ×[Z] Y) ×[X] Xᵢ)` needed for gluing | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t'_fst_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd | begin
delta t',
simp only [category.assoc, pullback_symmetry_hom_comp_fst_assoc,
pullback_right_pullback_fst_iso_inv_snd_fst_assoc, pullback.lift_fst_assoc, t_fst_fst,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.t'_fst_fst_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_fst_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.fst ≫ pullback.snd | begin
delta t',
simp only [category.assoc, pullback_symmetry_hom_comp_fst_assoc,
pullback_right_pullback_fst_iso_inv_snd_fst_assoc, pullback.lift_fst_assoc, t_fst_snd,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.t'_fst_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.snd | begin
delta t',
simp only [category.comp_id, category.assoc, pullback_symmetry_hom_comp_fst_assoc,
pullback_right_pullback_fst_iso_inv_snd_snd, pullback.lift_snd,
pullback_right_pullback_fst_iso_hom_snd],
end | lemma | algebraic_geometry.Scheme.pullback.t'_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_snd_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd | begin
delta t',
simp only [category.assoc, pullback_symmetry_hom_comp_snd_assoc,
pullback_right_pullback_fst_iso_inv_fst_assoc, pullback.lift_fst_assoc, t_fst_fst,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.t'_snd_fst_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_snd_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.fst ≫ pullback.snd | begin
delta t',
simp only [category.assoc, pullback_symmetry_hom_comp_snd_assoc,
pullback_right_pullback_fst_iso_inv_fst_assoc, pullback.lift_fst_assoc, t_fst_snd,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.t'_snd_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t'_snd_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst ≫ pullback.fst | begin
delta t',
simp only [category.assoc, pullback_symmetry_hom_comp_snd_assoc,
pullback_right_pullback_fst_iso_inv_fst_assoc, pullback.lift_fst_assoc, t_snd,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.t'_snd_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_fst_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.fst ≫
pullback.fst = pullback.fst ≫ pullback.fst ≫ pullback.fst | by simp only [t'_fst_fst_fst, t'_fst_snd, t'_snd_snd] | lemma | algebraic_geometry.Scheme.pullback.cocycle_fst_fst_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_fst_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.fst ≫
pullback.snd = pullback.fst ≫ pullback.fst ≫ pullback.snd | by simp only [t'_fst_fst_snd] | lemma | algebraic_geometry.Scheme.pullback.cocycle_fst_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.snd | by simp only [t'_fst_snd, t'_snd_snd, t'_fst_fst_fst] | lemma | algebraic_geometry.Scheme.pullback.cocycle_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_snd_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫
pullback.fst = pullback.snd ≫ pullback.fst ≫ pullback.fst | begin
rw ← cancel_mono (𝒰.map i),
simp only [pullback.condition_assoc, t'_snd_fst_fst, t'_fst_snd, t'_snd_snd]
end | lemma | algebraic_geometry.Scheme.pullback.cocycle_snd_fst_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_snd_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫
pullback.snd = pullback.snd ≫ pullback.fst ≫ pullback.snd | by simp only [pullback.condition_assoc, t'_snd_fst_snd] | lemma | algebraic_geometry.Scheme.pullback.cocycle_snd_fst_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle_snd_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.snd =
pullback.snd ≫ pullback.snd | by simp only [t'_snd_snd, t'_fst_fst_fst, t'_fst_snd] | lemma | algebraic_geometry.Scheme.pullback.cocycle_snd_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocycle (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j = 𝟙 _ | begin
apply pullback.hom_ext; rw category.id_comp,
{ apply pullback.hom_ext,
{ apply pullback.hom_ext,
{ simp_rw category.assoc,
exact cocycle_fst_fst_fst 𝒰 f g i j k },
{ simp_rw category.assoc,
exact cocycle_fst_fst_snd 𝒰 f g i j k } },
{ simp_rw category.assoc,
exact c... | lemma | algebraic_geometry.Scheme.pullback.cocycle | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gluing : Scheme.glue_data.{u} | { J := 𝒰.J,
U := λ i, pullback ((𝒰.map i) ≫ f) g,
V := λ ⟨i, j⟩, V 𝒰 f g i j, -- `p⁻¹(Uᵢ ∩ Uⱼ)` where `p : Uᵢ ×[Z] Y ⟶ Uᵢ ⟶ X`.
f := λ i j, pullback.fst,
f_id := λ i, infer_instance,
f_open := infer_instance,
t := λ i j, t 𝒰 f g i j,
t_id := λ i, t_id 𝒰 f g i,
t' := λ i j k, t' 𝒰 f g i j k,
t_fa... | def | algebraic_geometry.Scheme.pullback.gluing | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | Given `Uᵢ ×[Z] Y`, this is the glued fibered product `X ×[Z] Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
p1 : (gluing 𝒰 f g).glued ⟶ X | begin
fapply multicoequalizer.desc,
exact λ i, pullback.fst ≫ 𝒰.map i,
rintro ⟨i, j⟩,
change pullback.fst ≫ _ ≫ 𝒰.map i = (_ ≫ _) ≫ _ ≫ 𝒰.map j,
rw pullback.condition,
rw ← category.assoc,
congr' 1,
rw category.assoc,
exact (t_fst_fst _ _ _ _ _).symm
end | def | algebraic_geometry.Scheme.pullback.p1 | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The first projection from the glued scheme into `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
p2 : (gluing 𝒰 f g).glued ⟶ Y | begin
fapply multicoequalizer.desc,
exact λ i, pullback.snd,
rintro ⟨i, j⟩,
change pullback.fst ≫ _ = (_ ≫ _) ≫ _,
rw category.assoc,
exact (t_fst_snd _ _ _ _ _).symm
end | def | algebraic_geometry.Scheme.pullback.p2 | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The second projection from the glued scheme into `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
p_comm : p1 𝒰 f g ≫ f = p2 𝒰 f g ≫ g | begin
apply multicoequalizer.hom_ext,
intro i,
erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],
rw [category.assoc, pullback.condition]
end | lemma | algebraic_geometry.Scheme.pullback.p_comm | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_lift_pullback_map (i j : 𝒰.J) :
pullback ((𝒰.pullback_cover s.fst).map i) ((𝒰.pullback_cover s.fst).map j) ⟶
(gluing 𝒰 f g).V ⟨i, j⟩ | begin
change pullback pullback.fst pullback.fst ⟶ pullback _ _,
refine (pullback_right_pullback_fst_iso _ _ _).hom ≫ _,
refine pullback.map _ _ _ _ _ (𝟙 _) (𝟙 _) _ _,
{ exact (pullback_symmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition },
{ simpa using pullba... | def | algebraic_geometry.Scheme.pullback.glued_lift_pullback_map | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | (Implementation)
The canonical map `(s.X ×[X] Uᵢ) ×[s.X] (s.X ×[X] Uⱼ) ⟶ (Uᵢ ×[Z] Y) ×[X] Uⱼ`
This is used in `glued_lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued_lift_pullback_map_fst (i j : 𝒰.J) :
glued_lift_pullback_map 𝒰 f g s i j ≫ pullback.fst = pullback.fst ≫
(pullback_symmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition | begin
delta glued_lift_pullback_map,
simp only [category.assoc, id.def, pullback.lift_fst,
pullback_right_pullback_fst_iso_hom_fst_assoc],
end | lemma | algebraic_geometry.Scheme.pullback.glued_lift_pullback_map_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_lift_pullback_map_snd (i j : 𝒰.J) :
glued_lift_pullback_map 𝒰 f g s i j ≫ pullback.snd = pullback.snd ≫ pullback.snd | begin
delta glued_lift_pullback_map,
simp only [category.assoc, category.comp_id, id.def, pullback.lift_snd,
pullback_right_pullback_fst_iso_hom_snd],
end | lemma | algebraic_geometry.Scheme.pullback.glued_lift_pullback_map_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_lift : s.X ⟶ (gluing 𝒰 f g).glued | begin
fapply (𝒰.pullback_cover s.fst).glue_morphisms,
{ exact λ i, (pullback_symmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition ≫
(gluing 𝒰 f g).ι i },
intros i j,
rw ← glued_lift_pullback_map_fst_assoc,
have : _ = pullback.fst ≫ _ := (gluing 𝒰 f g).gl... | def | algebraic_geometry.Scheme.pullback.glued_lift | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The lifted map `s.X ⟶ (gluing 𝒰 f g).glued` in order to show that `(gluing 𝒰 f g).glued` is
indeed the pullback.
Given a pullback cone `s`, we have the maps `s.fst ⁻¹' Uᵢ ⟶ Uᵢ` and
`s.fst ⁻¹' Uᵢ ⟶ s.X ⟶ Y` that we may lift to a map `s.fst ⁻¹' Uᵢ ⟶ Uᵢ ×[Z] Y`.
to glue these into a map `s.X ⟶ Uᵢ ×[Z] Y`, we need to s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued_lift_p1 : glued_lift 𝒰 f g s ≫ p1 𝒰 f g = s.fst | begin
rw ← cancel_epi (𝒰.pullback_cover s.fst).from_glued,
apply multicoequalizer.hom_ext,
intro b,
erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],
delta glued_lift,
simp_rw ← category.assoc,
rw (𝒰.pullback_cover s.fst).ι_glue_morphisms,
simp_rw category.assoc,
erw [multicoequali... | lemma | algebraic_geometry.Scheme.pullback.glued_lift_p1 | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_lift_p2 : glued_lift 𝒰 f g s ≫ p2 𝒰 f g = s.snd | begin
rw ← cancel_epi (𝒰.pullback_cover s.fst).from_glued,
apply multicoequalizer.hom_ext,
intro b,
erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],
delta glued_lift,
simp_rw ← category.assoc,
rw (𝒰.pullback_cover s.fst).ι_glue_morphisms,
simp_rw category.assoc,
erw [multicoequali... | lemma | algebraic_geometry.Scheme.pullback.glued_lift_p2 | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_ι_to_V (i j : 𝒰.J) :
pullback (pullback.fst : pullback (p1 𝒰 f g) (𝒰.map i) ⟶ _) ((gluing 𝒰 f g).ι j) ⟶
V 𝒰 f g j i | (pullback_symmetry _ _ ≪≫
(pullback_right_pullback_fst_iso (p1 𝒰 f g) (𝒰.map i) _)).hom ≫
(pullback.congr_hom (multicoequalizer.π_desc _ _ _ _ _) rfl).hom | def | algebraic_geometry.Scheme.pullback.pullback_fst_ι_to_V | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | (Implementation)
The canonical map `(W ×[X] Uᵢ) ×[W] (Uⱼ ×[Z] Y) ⟶ (Uⱼ ×[Z] Y) ×[X] Uᵢ = V j i` where `W` is
the glued fibred product.
This is used in `lift_comp_ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_fst_ι_to_V_fst (i j : 𝒰.J) :
pullback_fst_ι_to_V 𝒰 f g i j ≫ pullback.fst = pullback.snd | begin
delta pullback_fst_ι_to_V,
simp only [iso.trans_hom, pullback.congr_hom_hom, category.assoc, pullback.lift_fst,
category.comp_id, pullback_right_pullback_fst_iso_hom_fst, pullback_symmetry_hom_comp_fst],
end | lemma | algebraic_geometry.Scheme.pullback.pullback_fst_ι_to_V_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_ι_to_V_snd (i j : 𝒰.J) :
pullback_fst_ι_to_V 𝒰 f g i j ≫ pullback.snd = pullback.fst ≫ pullback.snd | begin
delta pullback_fst_ι_to_V,
simp only [iso.trans_hom, pullback.congr_hom_hom, category.assoc, pullback.lift_snd,
category.comp_id, pullback_right_pullback_fst_iso_hom_snd, pullback_symmetry_hom_comp_snd_assoc]
end | lemma | algebraic_geometry.Scheme.pullback.pullback_fst_ι_to_V_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_ι (i : 𝒰.J) : pullback.lift pullback.snd (pullback.fst ≫ p2 𝒰 f g)
(by rw [← pullback.condition_assoc, category.assoc, p_comm]) ≫
(gluing 𝒰 f g).ι i = (pullback.fst : pullback (p1 𝒰 f g) (𝒰.map i) ⟶ _) | begin
apply ((gluing 𝒰 f g).open_cover.pullback_cover pullback.fst).hom_ext,
intro j,
dsimp only [open_cover.pullback_cover],
transitivity pullback_fst_ι_to_V 𝒰 f g i j ≫ fV 𝒰 f g j i ≫ (gluing 𝒰 f g).ι _,
{ rw ← (show _ = fV 𝒰 f g j i ≫ _, from (gluing 𝒰 f g).glue_condition j i),
simp_rw ← category... | lemma | algebraic_geometry.Scheme.pullback.lift_comp_ι | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"hom_ext"
] | We show that the map `W ×[X] Uᵢ ⟶ Uᵢ ×[Z] Y ⟶ W` is the first projection, where the
first map is given by the lift of `W ×[X] Uᵢ ⟶ Uᵢ` and `W ×[X] Uᵢ ⟶ W ⟶ Y`.
It suffices to show that the two map agrees when restricted onto `Uⱼ ×[Z] Y`. In this case,
both maps factor through `V j i` via `pullback_fst_ι_to_V` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_p1_iso (i : 𝒰.J) :
pullback (p1 𝒰 f g) (𝒰.map i) ≅ pullback (𝒰.map i ≫ f) g | begin
fsplit,
exact pullback.lift pullback.snd (pullback.fst ≫ p2 𝒰 f g)
(by rw [← pullback.condition_assoc, category.assoc, p_comm]),
refine pullback.lift ((gluing 𝒰 f g).ι i) pullback.fst
(by erw multicoequalizer.π_desc),
{ apply pullback.hom_ext,
{ simpa using lift_comp_ι 𝒰 f g i },
{ simp... | def | algebraic_geometry.Scheme.pullback.pullback_p1_iso | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | The canonical isomorphism between `W ×[X] Uᵢ` and `Uᵢ ×[X] Y`. That is, the preimage of `Uᵢ` in
`W` along `p1` is indeed `Uᵢ ×[X] Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_p1_iso_hom_fst (i : 𝒰.J) :
(pullback_p1_iso 𝒰 f g i).hom ≫ pullback.fst = pullback.snd | by { delta pullback_p1_iso, simp only [pullback.lift_fst] } | lemma | algebraic_geometry.Scheme.pullback.pullback_p1_iso_hom_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_p1_iso_hom_snd (i : 𝒰.J) :
(pullback_p1_iso 𝒰 f g i).hom ≫ pullback.snd = pullback.fst ≫ p2 𝒰 f g | by { delta pullback_p1_iso, simp only [pullback.lift_snd] } | lemma | algebraic_geometry.Scheme.pullback.pullback_p1_iso_hom_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_p1_iso_inv_fst (i : 𝒰.J) :
(pullback_p1_iso 𝒰 f g i).inv ≫ pullback.fst = (gluing 𝒰 f g).ι i | by { delta pullback_p1_iso, simp only [pullback.lift_fst] } | lemma | algebraic_geometry.Scheme.pullback.pullback_p1_iso_inv_fst | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_p1_iso_inv_snd (i : 𝒰.J) :
(pullback_p1_iso 𝒰 f g i).inv ≫ pullback.snd = pullback.fst | by { delta pullback_p1_iso, simp only [pullback.lift_snd] } | lemma | algebraic_geometry.Scheme.pullback.pullback_p1_iso_inv_snd | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_p1_iso_hom_ι (i : 𝒰.J) :
(pullback_p1_iso 𝒰 f g i).hom ≫ (gluing 𝒰 f g).ι i = pullback.fst | by rw [← pullback_p1_iso_inv_fst, iso.hom_inv_id_assoc] | lemma | algebraic_geometry.Scheme.pullback.pullback_p1_iso_hom_ι | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_is_limit : is_limit (pullback_cone.mk _ _ (p_comm 𝒰 f g)) | begin
apply pullback_cone.is_limit_aux',
intro s,
refine ⟨glued_lift 𝒰 f g s, glued_lift_p1 𝒰 f g s, glued_lift_p2 𝒰 f g s, _⟩,
intros m h₁ h₂,
change m ≫ p1 𝒰 f g = _ at h₁,
change m ≫ p2 𝒰 f g = _ at h₂,
apply (𝒰.pullback_cover s.fst).hom_ext,
intro i,
rw open_cover.pullback_cover_map,
have ... | def | algebraic_geometry.Scheme.pullback.glued_is_limit | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"hom_ext"
] | The glued scheme (`(gluing 𝒰 f g).glued`) is indeed the pullback of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_cover : has_pullback f g | ⟨⟨⟨_, glued_is_limit 𝒰 f g⟩⟩⟩ | lemma | algebraic_geometry.Scheme.pullback.has_pullback_of_cover | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_has_pullback {A B C : CommRing}
(f : Spec.obj (opposite.op A) ⟶ Spec.obj (opposite.op C))
(g : Spec.obj (opposite.op B) ⟶ Spec.obj (opposite.op C)) : has_pullback f g | begin
rw [← Spec.image_preimage f, ← Spec.image_preimage g],
exact ⟨⟨⟨_,is_limit_of_has_pullback_of_preserves_limit
Spec (Spec.preimage f) (Spec.preimage g)⟩⟩⟩
end | instance | algebraic_geometry.Scheme.pullback.affine_has_pullback | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"CommRing",
"opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_affine_has_pullback {B C : CommRing} {X : Scheme}
(f : X ⟶ Spec.obj (opposite.op C))
(g : Spec.obj (opposite.op B) ⟶ Spec.obj (opposite.op C)) : has_pullback f g | has_pullback_of_cover X.affine_cover f g | lemma | algebraic_geometry.Scheme.pullback.affine_affine_has_pullback | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"CommRing",
"opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_affine_has_pullback {C : CommRing} {X Y : Scheme}
(f : X ⟶ Spec.obj (opposite.op C))
(g : Y ⟶ Spec.obj (opposite.op C)) : has_pullback f g | @@has_pullback_symmetry _ _ _
(@@has_pullback_of_cover Y.affine_cover g f
(λ i, @@has_pullback_symmetry _ _ _ $ affine_affine_has_pullback _ _)) | instance | algebraic_geometry.Scheme.pullback.base_affine_has_pullback | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"CommRing",
"opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_affine_comp_pullback_has_pullback {X Y Z : Scheme}
(f : X ⟶ Z) (g : Y ⟶ Z) (i : Z.affine_cover.J) :
has_pullback ((Z.affine_cover.pullback_cover f).map i ≫ f) g | begin
let Xᵢ := pullback f (Z.affine_cover.map i),
let Yᵢ := pullback g (Z.affine_cover.map i),
let W := pullback (pullback.snd : Yᵢ ⟶ _) (pullback.snd : Xᵢ ⟶ _),
have := big_square_is_pullback (pullback.fst : W ⟶ _) (pullback.fst : Yᵢ ⟶ _)
(pullback.snd : Xᵢ ⟶ _) (Z.affine_cover.map i) pullback.snd pullbac... | instance | algebraic_geometry.Scheme.pullback.left_affine_comp_pullback_has_pullback | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover_of_left (𝒰 : open_cover X) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) | begin
fapply ((gluing 𝒰 f g).open_cover.pushforward_iso
(limit.iso_limit_cone ⟨_, glued_is_limit 𝒰 f g⟩).inv).copy 𝒰.J
(λ i, pullback (𝒰.map i ≫ f) g)
(λ i, pullback.map _ _ _ _ (𝒰.map i) (𝟙 _) (𝟙 _) (category.comp_id _) (by simp))
(equiv.refl 𝒰.J) (λ _, iso.refl _),
rintro (i : 𝒰.J),
cha... | def | algebraic_geometry.Scheme.pullback.open_cover_of_left | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"equiv.refl"
] | Given an open cover `{ Xᵢ }` of `X`, then `X ×[Z] Y` is covered by `Xᵢ ×[Z] Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover_of_right (𝒰 : open_cover Y) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) | begin
fapply ((open_cover_of_left 𝒰 g f).pushforward_iso (pullback_symmetry _ _).hom).copy 𝒰.J
(λ i, pullback f (𝒰.map i ≫ g))
(λ i, pullback.map _ _ _ _ (𝟙 _) (𝒰.map i) (𝟙 _) (by simp) (category.comp_id _))
(equiv.refl _) (λ i, pullback_symmetry _ _),
intro i,
dsimp [open_cover.bind],
apply p... | def | algebraic_geometry.Scheme.pullback.open_cover_of_right | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"equiv.refl"
] | Given an open cover `{ Yᵢ }` of `Y`, then `X ×[Z] Y` is covered by `X ×[Z] Yᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover_of_left_right (𝒰X : X.open_cover) (𝒰Y : Y.open_cover)
(f : X ⟶ Z) (g : Y ⟶ Z) : (pullback f g).open_cover | begin
fapply ((open_cover_of_left 𝒰X f g).bind (λ x, open_cover_of_right 𝒰Y (𝒰X.map x ≫ f) g)).copy
(𝒰X.J × 𝒰Y.J)
(λ ij, pullback (𝒰X.map ij.1 ≫ f) (𝒰Y.map ij.2 ≫ g))
(λ ij, pullback.map _ _ _ _ (𝒰X.map ij.1) (𝒰Y.map ij.2) (𝟙 _)
(category.comp_id _) (category.comp_id _))
(equiv.sigma_e... | def | algebraic_geometry.Scheme.pullback.open_cover_of_left_right | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"equiv.sigma_equiv_prod"
] | Given an open cover `{ Xᵢ }` of `X` and an open cover `{ Yⱼ }` of `Y`, then
`X ×[Z] Y` is covered by `Xᵢ ×[Z] Yⱼ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover_of_base' (𝒰 : open_cover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) | begin
apply (open_cover_of_left (𝒰.pullback_cover f) f g).bind,
intro i,
let Xᵢ := pullback f (𝒰.map i),
let Yᵢ := pullback g (𝒰.map i),
let W := pullback (pullback.snd : Yᵢ ⟶ _) (pullback.snd : Xᵢ ⟶ _),
have := big_square_is_pullback (pullback.fst : W ⟶ _) (pullback.fst : Yᵢ ⟶ _)
(pullback.snd : Xᵢ ... | def | algebraic_geometry.Scheme.pullback.open_cover_of_base' | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [] | (Implementation). Use `open_cover_of_base` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover_of_base (𝒰 : open_cover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) | begin
apply (open_cover_of_base' 𝒰 f g).copy
𝒰.J
(λ i, pullback (pullback.snd : pullback f (𝒰.map i) ⟶ _)
(pullback.snd : pullback g (𝒰.map i) ⟶ _))
(λ i, pullback.map _ _ _ _ pullback.fst pullback.fst (𝒰.map i)
pullback.condition.symm pullback.condition.symm)
((equiv.prod_punit 𝒰.J)... | def | algebraic_geometry.Scheme.pullback.open_cover_of_base | algebraic_geometry | src/algebraic_geometry/pullbacks.lean | [
"algebraic_geometry.gluing",
"category_theory.limits.opposites",
"algebraic_geometry.AffineScheme",
"category_theory.limits.shapes.diagonal"
] | [
"equiv.prod_punit",
"equiv.sigma_equiv_prod"
] | Given an open cover `{ Zᵢ }` of `Z`, then `X ×[Z] Y` is covered by `Xᵢ ×[Zᵢ] Yᵢ`, where
`Xᵢ = X ×[Z] Zᵢ` and `Yᵢ = Y ×[Z] Zᵢ` is the preimage of `Zᵢ` in `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
RingedSpace : Type* | SheafedSpace CommRing | abbreviation | algebraic_geometry.RingedSpace | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"CommRing"
] | The type of Ringed spaces, as an abbreviation for `SheafedSpace CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_res_of_is_unit_germ (U : opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : is_unit (X.presheaf.germ x f)) :
∃ (V : opens X) (i : V ⟶ U) (hxV : x.1 ∈ V), is_unit (X.presheaf.map i.op f) | begin
obtain ⟨g', heq⟩ := h.exists_right_inv,
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g',
let W := U ⊓ V,
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩,
erw [← X.presheaf.germ_res_apply (opens.inf_le_left U V) ⟨x.1, hxW⟩ f,
← X.presheaf.germ_res_apply (opens.inf_le_right U V) ⟨x.1, hxW⟩ g,
← ring_hom.... | lemma | algebraic_geometry.RingedSpace.is_unit_res_of_is_unit_germ | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit",
"is_unit_of_mul_eq_one",
"ring_hom.map_mul",
"ring_hom.map_one"
] | If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_of_is_unit_germ (U : opens X) (f : X.presheaf.obj (op U))
(h : ∀ x : U, is_unit (X.presheaf.germ x f)) :
is_unit f | begin
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`.
choose V iVU m h_unit using λ x : U, X.is_unit_res_of_is_unit_germ U f x (h x),
have hcover : U ≤ supr V,
{ intros x hxU,
rw [opens.mem_supr],
exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ },
-- Let `g x` denote the inverse of `... | lemma | algebraic_geometry.RingedSpace.is_unit_of_is_unit_germ | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit",
"is_unit.exists_right_inv",
"is_unit.mul_right_inj",
"is_unit_of_mul_eq_one",
"ring_hom.map_mul",
"ring_hom.map_one",
"supr"
] | If a section `f` is a unit in each stalk, `f` must be a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basic_open {U : opens X} (f : X.presheaf.obj (op U)) : opens X | { carrier := coe '' { x : U | is_unit (X.presheaf.germ x f) },
is_open' := begin
rw is_open_iff_forall_mem_open,
rintros _ ⟨x, hx, rfl⟩,
obtain ⟨V, i, hxV, hf⟩ := X.is_unit_res_of_is_unit_germ U f x hx,
use V.1,
refine ⟨_, V.2, hxV⟩,
intros y hy,
use (⟨y, i.le hy⟩ : U),
rw set.mem_set_... | def | algebraic_geometry.RingedSpace.basic_open | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_open_iff_forall_mem_open",
"is_unit",
"ring_hom.is_unit_map"
] | The basic open of a section `f` is the set of all points `x`, such that the germ of `f` at
`x` is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_basic_open {U : opens X} (f : X.presheaf.obj (op U)) (x : U) :
↑x ∈ X.basic_open f ↔ is_unit (X.presheaf.germ x f) | begin
split,
{ rintro ⟨x, hx, a⟩, cases subtype.eq a, exact hx },
{ intro h, exact ⟨x, h, rfl⟩ },
end | lemma | algebraic_geometry.RingedSpace.mem_basic_open | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top_basic_open (f : X.presheaf.obj (op ⊤)) (x : X) :
x ∈ X.basic_open f ↔ is_unit (X.presheaf.germ ⟨x, show x ∈ (⊤ : opens X), by trivial⟩ f) | mem_basic_open X f ⟨x, _⟩ | lemma | algebraic_geometry.RingedSpace.mem_top_basic_open | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_le {U : opens X} (f : X.presheaf.obj (op U)) : X.basic_open f ≤ U | by { rintros _ ⟨x, hx, rfl⟩, exact x.2 } | lemma | algebraic_geometry.RingedSpace.basic_open_le | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_res_basic_open {U : opens X} (f : X.presheaf.obj (op U)) :
is_unit (X.presheaf.map (@hom_of_le (opens X) _ _ _ (X.basic_open_le f)).op f) | begin
apply is_unit_of_is_unit_germ,
rintro ⟨_, ⟨x, hx, rfl⟩⟩,
convert hx,
rw germ_res_apply,
refl,
end | lemma | algebraic_geometry.RingedSpace.is_unit_res_basic_open | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit"
] | The restriction of a section `f` to the basic open of `f` is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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